Geodesics on an ellipsoid.pdf

Geodesics on an ellipsoidN E A geodesic on an oblate ellipsoid The study of geodesics on an ellipsoid arose in connection with geodesy specifically with the solution of triangulation networks. The figure of the Earth is well approximated by an oblate ellipsoid, a slightly flattened sphere. A geodesic is the shortest path between two points Isaac Newton on a curved surface, i.e., the analogue of a straight line on a plane surface. The solution of a triangulation network on an ellipsoid is therefore a set of exercises in spheroidal the great-circle distance using the mean Earth radius— trigonometry (Euler 1755). the relative error is less than 1%. However, the course If the Earth is treated as a sphere, the geodesics are great of the geodesic can differ dramatically from that of the circles (all of which are closed) and the problems re- great circle. As an extreme example, consider two points duce to ones in spherical trigonometry. However, Newton on the equator with a longitude difference of 179°59′; (1687) showed that the effect of the rotation of the Earth while the connecting great circle follows the equator, the results in its resembling a slightly oblate ellipsoid and, shortest geodesics pass within 180 km of either pole (the in this case, the equator and the meridians are the only flattening makes two symmetric paths passing close to the closed geodesics. Furthermore, the shortest path between poles shorter than the route along the equator). two points on the equator does not necessarily run along Aside from their use in geodesy and related fields such the equator. Finally, if the ellipsoid is further perturbed as navigation, terrestrial geodesics arise in the study of to become a triaxial ellipsoid (with three distinct semi- the propagation of signals which are confined (approxiaxes), then only three geodesics are closed and one of mately) to the surface of the Earth, for example, sound these is unstable. waves in the ocean (Munk & Forbes 1989) and the radio The problems in geodesy are usually reduced to two main cases: the direct problem, given a starting point and an initial heading, find the position after traveling a certain distance along the geodesic; and the inverse problem, given two points on the ellipsoid find the connecting geodesic and hence the shortest distance between them. Because the flattening of the Earth is small, the geodesic distance between two points on the Earth is well approximated by signals from lightning (Casper & Bent 1991). Geodesics are used to define some maritime boundaries, which in turn determine the allocation of valuable resources as such oil and mineral rights. Ellipsoidal geodesics also arise in other applications; for example, the propagation of radio waves along the fuselage of an aircraft, which can be roughly modeled as a prolate (elongated) ellipsoid (Kim & Burnside 1986). 1 2 1 GEODESICS ON AN ELLIPSOID OF REVOLUTION (the term spheroid is also used) was a well-accepted approximation to the figure of the Earth. The adjustment of triangulation networks entailed reducing all the measurements to a reference ellipsoid and solving the resulting two-dimensional problem as an exercise in spheroidal trigonometry (Bomford 1952, Chap. 3). N λ12 α2 α1 α0 E A φ1 F B s12 φ2 H Fig. 1. A geodesic AB on an ellipsoid of revolution. N is the north pole and EFH lie on the equator. Leonhard Euler It is possible to reduce the various geodesic problems into one of two types. Consider two points: A at latitude φ1 and longitude λ1 and B at latitude φ2 and longitude λ2 (see Fig. 1). The connecting geodesic (from A to B) is AB, of length s12 , which has azimuths α1 and α2 at the two endpoints.[1] The two geodesic problems usually considered are: Geodesics are an important intrinsic characteristic of curved surfaces. The sequence of progressively more complex surfaces, the sphere, an ellipsoid of revolution, and a triaxial ellipsoid, provide a useful family of surfaces for investigating the general theory of surfaces. Indeed, Gauss’s work on the survey of Hanover, which in1. the direct geodesic problem or first geodesic problem, volved geodesics on an oblate ellipsoid, was a key motigiven A, α1 , and s12 , determine B and α2 ; vation for his study of surfaces (Gauss 1828). Similarly, 2. the inverse geodesic problem or second geodesic the existence of three closed geodesics on a triaxial ellipproblem, given A and B, determine s12 , α1 , and α2 . soid turns out to be a general property of closed, simply connected surfaces; this was conjectured by Poincaré (1905) and proved by Lyusternik & Schnirelmann (1929) As can be seen from Fig. 1, these problems involve solving the triangle NAB given one angle, α1 for the direct (Klingenberg 1982, §3.7). problem and λ12 = λ2 − λ1 for the inverse problem, and its two adjacent sides. In the course of the 18th century these problems were elevated (especially in literature in 1 Geodesics on an ellipsoid of rev- the German language) to the principal geodesic problems (Hansen 1865, p. 69). olution For a sphere the solutions to these problems are simple exThere are several ways of defining geodesics (Hilbert & ercises in spherical trigonometry, whose solution is given Cohn-Vossen 1952, pp. 220–221). A simple definition by formulas for solving a spherical triangle. (See the aris as the shortest path between two points on a surface. ticle on great-circle navigation.) However, it is frequently more useful to define them as For an ellipsoid of revolution, the characteristic constant paths with zero geodesic curvature—i.e., the analogue of defining the geodesic was found by Clairaut (1735). A straight lines on a curved surface. This definition en- systematic solution for the paths of geodesics was given compasses geodesics traveling so far across the ellipsoid’s by Legendre (1806) and Oriani (1806) (and subsequent surface (somewhat less than half the circumference) that papers in 1808 and 1810). The full solution for the diother distinct routes require less distance. Locally, these rect problem (complete with computational tables and a geodesics are still identical to the shortest distance be- worked out example) is given by Bessel (1825).[2] tween two points. Much of the early work on these problems was carried out By the end of the 18th century, an ellipsoid of revolution by mathematicians—for example, Legendre, Bessel, and 1 Equations for a geodesic Barnaba Oriani Gauss—who were also heavily involved in the practical aspects of surveying. 1. for example (Hutton 1811). and it is therefore the proper itinerary measure of the distance between those two points. the analysis of the stability of closed geodesics. while mathematicians pursued the solution of geodesics on a triaxial ellipsoid.[4] This section treats the problem on an ellipsoid of revolution (both oblate and prolate). This terminology was introduced into English either as “geodesic line” or as “geodetic line”. is called a geodetic or geodesic line: it has the property of being the shortest which can be drawn between its two extremities on the surface of the Earth.[3] The term “geodesic line” was Friedrich Bessel . was preferred. etc. or deduced from trigonometrical measures. some of these are described in the article on geographical distance. the disciplines diverged: those with an interest in geodesy concentrated on the practical aspects such as approximations suitable for field work. A line traced in the manner we have now been describing. During the 18th century geodesics were typically referred to as “shortest lines”. When determining distances on the earth. The problem on a triaxial ellipsoid is covered in the next section. Beginning in about 1830.1 Equations for a geodesic 3 coined by Laplace (1799b): Nous désignerons cette ligne sous le nom de ligne géodésique [We will call this line the geodesic line]. to “geodesic”. frequently shortened.1. by the means we have indicated. Alexis Clairaut In its adoption by other fields “geodesic line”. various approximate methods are frequently used. R = ν cosφ is the radius of the circle of latitude φ. a < b. Krakiwsky & Thomson (1974. β. Differential element of a geodesic on an ellipsoid. sin α ds = R dλ. λ1 ) and (φ2 . e2 . (1) gives a dλ Fig. these allow the geodesics of any length to be computed accurately. 4. then ds is related to dφ and dλ by . Rapp (1993. 2 and 3. Here the equations for a geodesic are developed. the eccentricity e2 = f(2 − f). A point P at latitude φ on the meridian (red) is mapped to a point which case f. R sin α = const. The geometric construction for parametric latitude. the ellipsoid is taken to be oblate. Define the flattening f = (a − β φ b)/a. Differential element of a meridian ellipse. R dλ ρ dφ N α λ2 s12 = L(ϕ. in Fig. §10). The shortest path or geodesic entails finding that function φ(λ) which minimizes s12 . 3. ϕ′ ) dλ. (In most applications in geodesy. a sinβ b Z P′ P Consider an ellipsoid of revolution with equatorial radius a and polar semi-axis b.[5] Differentiating this relation and manipulating the result gives (Jekeli 2012. λ1 where φ is a function of λ satisfying φ(λ1 ) = φ1 and φ(λ2 ) = φ2 . λ2 ) is given by φ 0 R a ∫ Fig.) P′ on a sphere of radius a (shown as a blue circle) by keeping the Let an elementary segment of a path on the ellipsoid have radius R constant. dφ where φ′ = dφ/dλ and L depends on φ through ρ(φ) and R(φ). The elementary segment is therefore given by Clairaut (1735) first found this relation. length ds. and the second ec0 R = a cosβ a centricity e′ = e/(1 − f). and ν is the normal radius of curvature. From Figs. ∂ϕ′ Substituting for L and using Eqs. Bagratuni (1962. (2. Eq. §4). §1. and e′2 are negative. ϕ′ ) dλ. and Borre & Strang (2012) also provide derivations of these equations.4 1 GEODESICS ON AN ELLIPSOID OF REVOLUTION -dR b ds2 = ρ2 dϕ2 + R2 dλ2 ρ dφ ρ φ or √ ds = ρ2 ϕ′2 + R2 dλ ≡ L(ϕ. a > b. This is an exercise in the calculus of variations and the minimizing condition is given by the Beltrami identity. where ρ is the meridional radius of curvature. ds φ + dφ φ L − ϕ′ ∂L = const. we see that if its azimuth is α.95)) . the theory applies without change to prolate ellipsoids. cos α ds = ρ dϕ = −dR/ sin ϕ. a similar derivation is presented by Lyusternik (1964. The following derivation closely follows that of Bessel (1825).2). §15). however. The length of an arbitrary path between (φ1 . using a geometrical construction. 2. spherical longitude. a dσ dω sin ϕ Up to this point. 7). 5. R cos ϕ the spherical arc length.76)) . 7. (1). ds ρ dλ sin α = . β2 Geodesic problem mapped to the auxiliary N ω (λ . dϕ cos α = . the variables referred to the auxiliary sphere are shown with the corresponding quantities dα = sin ϕ dλ.[7] Bessel now specializes to an ellipsoid in which R and Z are related by β (φ) R2 Z2 + 2 = 1. (1) and (3) gives differential equations for s and λ H F α0 cos α dσ = dβ. the a = cos β = √1 − e2 sin2 ϕ . we obtain A = α1 . for the ellipsoid shown in parentheses. ds ν cos ϕ cosβ dω dα tan ϕ sin α = . This is the sine rule of spherical trigonometry relating two a2 b2 sides of the triangle NAB (see Fig. β. s and ω. leads to a system of ordinary which the geodesic crosses the equator in the northward differential equations for a geodesic (Borre & Strang direction. the point at This. it is useful to consider the triangle NEP representing a geodesic starting at the equator. sphere. we have not made use of the specific equations for an ellipsoid. see This relation between β and φ can be written as . Differentiating this and setting dR/dZ = −sinφ/cosφ gives Fig. Differential element of a geodesic on a sphere. 2012. A β1 B σ12 . together with Eqs. Combining Eqs. The elementary geodesic problem on the auxiliary sphere. 6. and included angle N = ω12 . R sin ϕ Z cos ϕ − = 0.71) and (11. using dβ N [6] α dσ β + dβ R = a cos β β dω (see Fig. is used as the origin for σ. If the side EP is extended by moving P infinitesimally (see Fig. we obtain N ω12 α2 α1 α0 E Fig. 1 ds dλ sin β = = .λ0) α P E σ (s) sin α dσ = cos β dω. (11. 6. and NB = ½π − β2 and their opposite angles B = π − α2 and eliminating Z from these equations. and Clairaut’s relation then becomes sin α1 cos β1 = sin α2 cos β2 . In this figure. ds ν We can express R in terms of the parametric latitude.1. 5). NA = ½π − β1 . 4 for the geometrical construction). 4).1 Equations for a geodesic 5 Fig. Fig. 2 a b G where Z is the height above the equator (see Fig. Eqs. Quantities without subscripts refer to the arbitrary point P. In order to find the relation for the third side AB = σ12 . and indeed the derivation applies to an arbitrary surface of revolution. E. 13 and 14). The quantities ω. we write dω = E sin α0 dσ. §2. ellipsoid. 0 1 + (1 − f ) 1 + k 2 siningσto k2 . π]. and the limits on the integral are chosen so that s(σ = 0) = 0. λ can be reduced to the conventional sin β = 1 − e2 cos2 β. ∫ .λ0 180 ∫ σ e2 o -45 dσ ′ √ = ω − sin α0 2 2 ′ 1 − e cos β(σ . yields see Figs. 180) pointed out that the equation for s is the same as the equation for the arc on an ellipse with semi-axes b(1 + e′2 cos2 α0 )1/2 and b. Furthermore. α0 ) = cos α0 sin σ.2 Behavior of geodesics √ σ 1 − e2 cos2 β(σ ′ . Once the √ problem is solved. however. However. 2 σ′ o 45 1 + sin ′φ o λ − λ0 = (1 − f ) sin α0 dσ 360 2 ′ o 2 0 0 1 − cos α0 sin σ o λ . cos2 β Fig. In using these integral relations. it should be viewed both of these differential equations and thereby to express merely as a convenient tool for solving for a particular s and λ as integrals. σ √ Geodesic on an oblate ellipsoid (f = 1/50) with α0 = 45°. Legendre (1811. §6).4). and s are likewise allowed to increase without limit.1. we allow σ to increase which is the normal definition of the parametric latitude continuously (not restricting it to a range [−π. parable in complexity to the method for the exact solution given above (Jekeli 2012. Applying the sine rule to the vertices geodesic. there are other closed geodesics. By this device a great circle can be mapped exactly to a geodesic on an ellipsoid of revolu√ tion. This (For the very flattened ellipsoids. of the spherical quantities depend on α0 . the mapping is a dσ dω not a consistent mapping of the surface of the sphere to The last step is to use σ as the independent parameter[8] in the ellipsoid or vice versa. resp. Meridians and the equator are the only closed geodesics. we have auxiliary sphere. Substituting this into the equation for ds/dσ and integrating the result gives 1. encircles the on an ellipsoid. (3) and Clairaut’s relation. αFig. range. 6 gives There are also several ways of approximating geodesics on an ellipsoid which usually apply for sufficiently short lines (Rapp 1991.6 1 GEODESICS ON AN ELLIPSOID OF REVOLUTION and the limits on the integrals are chosen so that λ = λ0 at the equator crossing. because the equations for s and λ in terms dλ 1 ds = = 1 − e2 cos2 β. p. N 0 where k = e′ cos α0 . Latitude as a function of longitude for a single cy∫ σ 2−f cle ofdσ the′ . these are typically comsin β = sin β(σ. instead. for example) as the great circle. geodesic from one northward equatorial cross√ = ω − f sin α0 2 ′ the next. 0 1+ 0 ) 9. 8. sin ϕ so that the differential equations for the geodesic become This completes the solution of the path of a geodesic using the auxiliary sphere. In order to express the equation for λ in terms of σ. resp. ∫ . which follows from Eq. E and G in the spherical triangle EGP in Fig. where α0 is the azimuth at E. σ = 0. λ. geodesic. √ tan β = 1 − e2 tan ϕ = (1 − f ) tan ϕ. α0 ) ′ s = dσ b 1−f ∫0 σ √ = 1 + k 2 sin2 σ ′ dσ ′ . expressed in terms of parametric latitude.) This follows from the equations for the geodesics given in the previous section. 8 shows the simple closed geodesics which consist of the meridians (green) and the equator (red). β. 10). N E E Fig. Two additional closed geodesics for the oblate ellipsoid. vertices. λ falls short of a full circuit of the equator by approximately 2π f sinα0 (for a prolate ellipsoid. jumping by π each time the geodesic crosses the pole. For nearly all values of α0 . (4). 9 to 11. 12. the longitude will vary the same way as for a sphere. 10. . (5) becomes λ = ω + λ0 . A full cycle of the geodesic. All other geodesics are typified by Figs. i. 11. 10. resp.e. the geodesic N will fill that portion of the ellipsoid between the two vertex latitudes (see Fig. while the longitude simplifies to λ = (1 − f)σ + λ0 . Thus. this quantity is negative and λ completes more that a full circuit. reduces to the length of an arc of an ellipse with semi-axes a and b (as expected). see Fig. the maximum length of a 5 circuits. Fig. resp. function of the longitude about the nodes. N For meridians. even.1. Following the geodesic on the ellipsoid for about vals of πb. their behavior. The distance reduces to the arc of a circle of radius b (and not a). φ = 0 on the ellipsoid) corresponds to α0 = ½π. Geodesic on a prolate ellipsoid (f = −1/50) with α0 = 45°. from one northward crossing of the equator to the next. Side view. The latitude is an odd. on each successive northward crossing of the equator (see Fig.. b/a = 2/7.2 Behavior of geodesics 7 intersection. E The equator (β = 0 on the auxiliary sphere. The geodesic completes one full oscillation in latitude before the longitude has increased by 360°. A geodesic that is nearly equatorial will intersect the equator at interFig. (Here the qualification “simple” means that the geodesic closes on itself without an intervening self- Fig.| = ±(½π − |α0 |). 12). The distance. it is worth reviewing Fig. The equatorial crossings are called nodes and the points of maximum or minimum latitude are called vertices. Eq. the vertex latitudes are given by |&beta. Figure 9 shows latitude as a function of longitude for a geodesic starting on the equator with α0 = 45°. is shown. we have α0 = 0 and Eq. As a consequence. the maximum length is πa). The same geodesic after about 70 circuits. 13. 11). Before solving for the geodesics. equatorial geodesic which is also a shortest path is πb on an oblate ellipsoid (on a prolate ellipsoid. s = bσ. Compare with Fig. λ. as . |f| ≤ 1/50. Such techniques can be used for arbitrary flattening f. It is also possible to evaluate the integrals (4) and (5) by numerical quadrature (Saito 1970) (Saito 1979) (Sjöberg & Shirazian 2012) or to apply numerical techniques for the solution of the ordinary differential equations. Mathar (2007) has tackled the more complex problem of geodesics on the surface at a constant altitude.1 mm for the WGS84 ellipsoid.192°). noting that the required derivative is just the integrand of the distance integral. (2) with [ρ. As a consequence. Eqs.5. they do not offer the speed and accuracy of the series expansions described above. Helmert (1880) gives a similar series. the evaluation of the integrals in terms of elliptic integrals (see below) also provides a fast and accurate solution. (4) and (5). In geodetic applications.. 13 and 14. it is typically only necessary to retain terms up to O(f J−1 ) in that integral. Series expansions for Bj can readily be found and the result truncated so that only terms which are O(f J ) and larger are retained. which may be found using Newton’s method. h. for this purpose. the second form of the longitude integral is preferred (since it avoids the near singular behavior of the first form when geodesics pass close to a pole). 75. (4) and (5). Top view. Karney (2013) gives expansions carried out for J = 6 which suffices to provide full double precision accuracy for |f| ≤ 1/50. Oriani (1833) instead uses series reversion so that σ can be found without iteration. this problem has a unique root. for the green (resp. Vincenty (1975a) instead relies on a simpler (but slower) function iteration to solve for σ.[10] The reverted series converges somewhat slower that the direct series and.) This prescription is followed by many authors (Legendre 1806) (Oriani 1806) (Bessel 1825) (Helmert 1880) (Rainsford 1955) (Rapp 1993). b/a < ½. §3. Since the integrand in the distance integral is positive. the integrand is an even periodic function of period π. Legendre (1811) writes the integrals. Furthermore. the integrals are typically evaluated as a series. A. Bj sin 2jσ j=1 where B0 = 1 + O(f) and Bj = O(f j ). Two such geodesics are illustrated in Figs. s. Furthermore. If the ellipsoid is sufficiently oblate. 3) complete oscillations about the equator on one circuit of the ellipsoid. 2013).8 1 GEODESICS ON AN ELLIPSOID OF REVOLUTION N Fig. Trigonometric series of this type can be conveniently summed using Clenshaw summation. (2) (Kivioja 1971) (Thomas & Featherstone 2005) (Panou et al. addenda) supplements the reverted series with one step of Newton’s method to maintain accuracy. 14. Here b/a = 2/7 and the equatorial azimuth. if f is small. ν + h].19). Legendre Geodesics on an ellipsoid was an early application of elliptic integrals. In both integrals. Eqs. M. it is necessary to find σ given s. 1. the term dependent on σ is multiplied by a small quantity k2 = O(f). where f is small. Vincenty (1975a) uses J = 3 which provides an accuracy of about 0. Karney (2013.e.175° (resp. and the longitude. e. However.3 Evaluation of the integrals Solving the geodesic problems entails evaluating the integrals for the distance. for arbitrary f. if |f| > 1/100. another class of simple closed geodesics is possible (Klingenberg 1982. blue) geodesic is chosen to be 53. Eqs. the integrals can both be written in the form I = B0 σ + ∞ ∑ In order to solve the direct geodesic problem.[9] (Because the longitude integral is multiplied by f. In particular. Eqs. above the ellipsoid by solving the corresponding ordinary differential equations. α0 .g.. so that the geodesic completes 2 (resp. ν] replaced by [ρ + h. On the other hand. i. [15] Use the distance and longitude equations. and s12 . and Π(φ. Intermediate points. cos σ = cos β cos ω = tan α0 cot α. The second formula in Eq.9). to solve the corresponding great-circle problem on the sphere. are incomplete the determination of s2 ) needs to be repeated for each elliptic integrals in the notation of DLMF (2010. we can apply Napier’s rules for quadrantal triangles to the triangle NEP on the auxiliary sphere which give Arthur Cayley sin α0 = sin α cos β = tan ω cot σ. k). α2 . k) + 1 − 2 Π(ϕ. (7) follows directly from the first form of Eq. Once the first waypoint is found. on a geodesic can be found by holding φ1 and α1 fixed and solving the direct problem for several values of s12 .7.4 Solution of the direct problem 9 §19.1. In implementing this program. α . E(φ. α2 . k) = dθ 1 − α2 sin2 θ 0 ( ) k2 k2 = 2 F (ϕ. Determine s2 = s1 + s12 and invert the distance equation to find σ2 . It is essentially the program laid out by Bessel (1825). sin β = cos α0 sin σ = cot α tan ω. Fast algorithms for computing elliptic integrals are given by Carlson (1995) in terms of symmetric elliptic integrals. For this purpose. due to Cayley (1870). Convert β2 to φ2 and substitute σ2 and ω2 into the longitude equation to give λ2 . . way-points. and α2 . k). b λ = (1 − f ) sin α0 G(σ.2(ii)). (7).[16] The overall method follows the procedure for solving the direct problem on a sphere. We are given φ1 . and s. λ. to find s1 and λ0 . . s = E(σ.4 Solution of the direct problem The basic strategy for solving the geodesic problems on the ellipsoid is to map the problem onto the auxiliary sphere by converting φ. Eq. −e′2 . and ω1 . §5. =χ− √ e sin α0 H(σ. k). α2 . 6 with P replaced by either A (subscript 1) or B (subscript 2). 1 + e′2 where √ tan χ = 1 + e′2 tan ω.[17] Helmert (1880. we will frequently need to solve the “elementary” spherical triangle for NEP in Fig. α2 . α α ∫ ϕ 2 cos θ √ H(ϕ. Equation (6) is conveniently inverted using Newton’s method. together with the known value of λ1 . The use of elliptic integrals provides a good method of solving the geodesic problem for |f| > 1/50. and to transfer the results back to the ellipsoid.8)). (19. is more convenient for calculation since the elliptic integral appears in a small term. (4) and (5). σ1 . k). Solve the triangle problem with P = B and α0 and σ2 given to find β2 . k) = dθ 2 2 2 0 (1 − α sin θ) 1 − k 2 sin θ ( ) 1 1 = 2 F (ϕ. cos2 α0 . Eqs. only the last portion of the solution (starting with and F(φ.[13] 1. new value of s12 . (5). From φ1 we obtain β1 (using the formula for the parametric latitude). ′2 ϕ We can also stipulate that cosβ ≥ 0 and cosα0 ≥ 0. ik). ik). cos α = cos ω cos α0 = cot σ tan β. ω and σ. ω2 . α α ∫ 2 sin ω = sin σ sin α = tan β tan α0 . k).[14] Implementing this plan for the direct problem is straightforward. and most subsequent authors.[11][12] The first formula for the longitude in Eq. ik) . k) + 1 − 2 Π(ϕ. to β. α1 . The equivalence of the two forms follows from DLMF (2010. We now solve the triangle problem with P = A and β1 and α1 given to find α0 . 1 + k 2 sin2 σ and √ 1 − k 2 sin2 θ G(ϕ. 17). on any surface. The red line is the cut locus. 0 ≤ λ12 ≤ π. This completes the solution. the locus of points which have multiple (two in this case) shortest geodesics from A. |ϕ2 | ≤ |ϕ1 | . Thus.5 Solution of the inverse problem The ease with which the direct problem can be solved results from the fact that given φ1 and α1 .. Then as α1 is varied between 0° and 180°. . Gauss (1828) showed that. given φ1 . Suppose point A in the inverse problem has φ1 = −30° and λ1 = 0°. (The flattening has been increased to 1/10 in order to accentuate the ellipsoidal effects. Solve the so-called hybrid geodesic problem. Geodesics from a single point (f = 1/10. it is worth understanding better the behavior of geodesics. and the cut locus. and α1 find λ12 .) o 90 1.₁₂| ≤ (1 − f)π. Compare the resulting λ12 with the desired value and adjust α1 until the two values agree. which are the geodesic circles centered A. Thus. geodesics and geodesic circle intersect at right angles. the longitude at which the geodesic intersects φ = φ2 varies between 0° and 180° (see Fig. corresponding to the first intersection of the geodesic with the circle φ = φ2 . it is a segment of the circle of latitude centered on the point antipodal to A. the cut locus is a segment of the anti-meridian centered on the point antipodal to A. Suppose φ2 = 20°. Eqs. 2. φ1 = −30°) E Fig. Fig. this determines α1 and s12 respectively. 3. the parameter in the distance and longitude integrals. The solution of the inverse problem involves determining. o ϕ1 ≤ 0. we can immediately find α0 . this relation is exact and as a consequence the equator is only a shortest geodesic if |&lambda.10 1 GEODESICS ON AN ELLIPSOID OF REVOLUTION 1. (4) and (5). This behavior holds provided that |&phi. φ = −φ1 .λ1 o 180 they cease to be shortest paths. λ12 as a function of α1 for φ1 = −30° and φ2 = 20°. The longitudinal extent of cut locus is approximately λ12 ∈ [π − f π cosφ1 . π + f π cosφ1 ]. This suggests the following strategy for solving the inverse problem (Karney 2013). φ1 = 0. we are given λ12 .. the cut locus is a point. Guess a value of α1 . λ12 (There is no loss of generality in this assumption. On a sphere. In the case of the inverse problem. 4. the solution of the problem requires that α0 be found iteratively. this time. keeping the starting point fixed and varying the azimuth. o 0 o 30 φ o o 30 o 60 α1 = 90 o 0 o o -30 120 o 150 o -60 o 0 o 45 o 90 o 180 o 135 λ . the inverse problem may be solved by determining the value α1 which results in the given value of λ12 when the geodesic intersects the circle φ = φ2 . 16. the green line in the figure. φ2 .) Also shown (in green) are curves of constant s12 . Assume that the points A and B satisfy Fig. 180 . geodesics which lie on a meridian or the equator. the ellipsoid has been “rolled out” onto a plate carrée projection. The geodesics shown on a plate carrée projection. 17. On an oblate ellipsoid (shown here). 15 shows geodesics (in blue) emanating A with α1 a multiple of 15° up to the point at which Each of these steps requires some discussion. In Fig. λ12 = π.₂| ≤ |φ1 | (otherwise the geodesic does not reach φ2 for some values of α1 ). and α2 . o 0 o 0 o 90 α1 o 180 Fig. but we cannot easily relate this to the equivalent spherical angle ω12 because α0 is unknown. If A lies on the equator. 15. Otherwise. and this means that meridional geodesics stop being shortest paths before the antipodal point is reached. geodesic circles. First treat the “easy” cases. Geodesics. 16. for a given point B with latitude φ2 and longitude λ2 which blue and green curves it lies on. since the symmetries of the problem can be used to generate any configuration of points from such configurations. For a prolate ellipsoid. Before tackling this. s12 . An alternative method for solving the inverse problem is given by Helmert (1880. For an oblate ellipsoid. The shortest geodesic follows the equator if φ1 = φ2 = 0 and |&lambda. cos β1 sin β2 − sin β1 cos β2 cos ω12 with ω12 = λ12 . There is no longitudinal restriction on equatorial geodesics. northward. this may not matter (depending on how step 4 is handled). 3. to find s12 and λ12 . Let us rewrite the Eq. ω1 . σ2 .1. (4) and (5). taking cosα2 ≥ 0 (corresponding to the first. σ2 . For a prolate ellipsoid. 2.[18] Now find α2 from sinα2 = sinα0 /cosβ2 .[19] 4. In a subsequent report.) • 2−f √ dσ ′ 2 σ1 1 + (1 − f ) 1 + k 2 sin σ ′ = ω12 − f sin α0 I(σ1 . If α1 = α2 . crossing of the circle φ = φ2 ). Next. but he is only partially successful—the NGS (2012) implementation still includes Vincenty’s fix still fails to converge in some cases. however. Any of a number of rootfinding algorithms can be used to solve such an equation. where g(0) ≤ 0 and g(π) ≥ 0.13). In fact. Here is a catalog of those cases: • φ1 = −φ2 (with neither point at a pole).5 Solution of the inverse problem 11 1. if B lies on the cut locus of A there are multiple azimuths which yield the same shortest distance. (5) as ∫ F. which requires a good starting guess. ϕ2 ) − λ12 = 0. α0 ). more seriously. Karney (2013) uses Newton’s method. σ2 is given by tanσ2 = tanβ2 /cosα2 and ω2 by tanω2 = tanσ2 /sinα0 . (This occurs when λ12 ≈ ±π for oblate ellipsoids. Rainsford (1955) advocates this method and Vincenty (1975a) adopted it in his solution of the inverse problem. and obtaining an updated estimate of ω12 using λ12 = ±π (with neither point at a pole). There are infinitely many geodesics which can be generated by varying . Helmert ω12 = λ12 + f sin α0 I(σ1 . The solution of the hybrid geodesic problem is as follows. φ1 . however it may be supplemented by a fail-safe method. σ1 . (This occurs when φ1 + φ2 ≈ 0 for prolate ellipsoids. use the distance and longitude equations. the geodesic is unique. The curve in Fig. α0 ). that the process fails to converge at all for nearly antipodal points. This fixed point iteration is repeated until convergence. there is a unique root in the interval α1 ∈ [0. In order to discuss how α1 is updated. However. φ2 ) where we regard φ1 and φ2 as parameters and α1 as the independent variable. π]. Vincenty (1975b) attempts to cure this defect. The shortest distance returned by the solution of the inverse problem is (obviously) uniquely defined. This may be a bad approximation if A and B are nearly antipodal (both the numerator and denominator in the formula above become small). Otherwise there are two geodesics and the second one is obtained by interchanging α1 and α2 . let us define the root-finding problem in more detail. Otherwise there are two geodesics and the second one is obtained by negating α1 and α2 . the geodesic is unique.) • A and B are at opposite poles. 17 shows λ12 (α1 . §5. If α1 = 0 or ±π. R. solving the resulting problem on auxiliary sphere. The drawbacks of this method are that convergence is slower than obtained using Newton’s method (as described above) and. Eqs. Lee (2011) has compared 17 methods for solving the inverse problem against the method given by Karney (2013). In most cases a suitable starting value of α1 is found by solving the spherical inverse problem[14] tan α1 = cos β2 sin ω12 . It starts the same way as the solution of the direct problem. λ12 = ω12 − f sin α0 σ2 Helmert’s method entails assuming that ω12 = λ12 . We seek the value of α1 which is the root of g(α1 ) ≡ λ12 (α1 . such as the bisection method. the meridian is no longer the shortest geodesic if λ12 = ±π and the points are close to antipodal (this will be discussed in the next section).[14] Finally.₁₂| ≤ (1 − f)π. ϕ1 . and λ0 . the shortest geodesic lies on a meridian if either point lies on a pole or if λ12 = 0 or ±π. solving the triangle NEP with P = A to find α0 . to guarantee convergence. 12 1 GEODESICS ON AN ELLIPSOID OF REVOLUTION the azimuths so as to keep α1 + α2 constant. Definition of reduced length and geodesic scale.6 Differential behavior of geodesics second geodesic a small distance t(s) away from it. This is useful antipodal. (For Various problems involving geodesics require knowing spheres. and a 1.) in trigonometric adjustments (Ehlert 1993). The solution dα1 m12dα1 B A dt1 A M12dt1 B Fig. s1 ) = 0. ds2 . . Consider a reference geodesic. s1 ) = 1. etc. may be expressed as the sum of two independent solutions t(s2 ) = Cm(s1 . Gauss (1828) showed that t(s) obeys the Gauss-Jacobi equation d2 t(s) = K(s)t(s). F. s2 ) C. s2 ) + DM (s1 . Gauss where m(s1 . 18. determining the physical properties of signals which follow geodesics. parameterized by s the length from the northward equator crossing. M (s1 . this prescription applies when A and B are their behavior when they are perturbed. where K(s) is the Gaussian curvature at s. s2 ) . dm(s1 . . = 1. . ds2 s2 =s1 . s2 ) . dM (s1 . . = 0. . The reduced length obeys a reciprocity relation. Christoffel (1869) made an extensive study of their properties. m12 + m21 = 0. lie on the same geodesic. Their derivatives are dm12 = M21 . . 2. the geodesic scale. and 3. Christoffel Assuming that points 1. then the following addition rules apply (Karney 2013). B. and M(s1 . s2 ) = m12 . ds2 m12 E.[20] Their basic definitions are illustrated in Fig. ds2 s2 =s1 We shall abbreviate m(s1 . 18. ds2 dM12 1 − M12 M21 =− . s2 ) = M 12 . the so-called reduced length. (46)). is given in − cos σ1 cos σ2 J(σ2 ) − J(σ1 ) . The coordinates of another point B are given by r = s12 and θ = ½π ∫ σ 2 k 2 sin σ ′ − α1 and these coordinates are used to find the projected √ dσ ′ J(σ) = 0 1 + k 2 sin2 σ ′ coordinates on a plane map. φ1 = α1 = 0. ik) − F (σ.7 Geodesic map projections 13 length s12 . The polar coordinate system (r. lines of constant x and lines of constant y intersect at right angles √ √ √ on the surface. (9) by a constant K. m12 /b = 1 + k 2 sin2 σ2 cos σ1 sin σ2 − 1 + k 2 sin2 σThe σ1 cos σ2required to solve the inverse method usderivative 1 sin ( ) ing Newton’s method. ional geodesic starting on the equator. K= (1 − e2 sin2 ϕ)2 b2 1 = = 4 . If A is on the equator and α1 = metric adjustments over small areas. The point at which m12 becomes zero is the point conjugate to the starting point. m13 = m12 M23 + m23 M21 . the Jacobi condition is a local property .5. The point (x. that there is no point conjugate to m23 A between A and B. In order for a geodesic between A and B. to be a shortest path it must satisfy the Jacobi condition (Jacobi 1837) (Jacobi 1866. φ2 ) / ∂α1 . for a meridgeodesic defined by A and α1 as the x axis. tion. On a closed surface such as lines of constant θ intersect at right angles on the surface. the basic properties of geodesics (Gauss 1828). √ 1 + k 2 sin2 σ2 M12 = cos σ1 cos σ2 + √ sin σ1 sin σ2 1 + k 2 sin2 σ1 ( ) 1. it may be possible ½π. ∂λ12 (α1 . In. lines of constant r and differing by dα1 is m12 dα1 . 1 + k 2 sin2 σ1 They are based on polar and rectangular geodesic coordiwhere nates on the surface (Gauss 1828). In trigonogeodesic a distance s32 = y. deed. these quantities the reference geodesic to an intermediate point C and oscillate with a period of about 2π in σ12 and grow linthen turning ½π counter-clockwise and traveling along a early with distance at a rate proportional to f. an ellipsoid. (6. M12 = cos( Ks12 ). we y) is found by traveling a distance s13 = x from A along have M 12 = cosσ12 . if α0 ≠ 0. If α1 to approximate K(s) in Eq.1. The latter condition above can be used to determine whether the shortest path is a meridian in the case of √ √ a prolate ellipsoid with |&lambda.₁₂| = π. if α0 = 0. Eq. As we see from Fig. the separation it is called the exponential map.The scale of the projection in the radial direction is unity. In the typical case. tion is unity.. In this = 0. this gives the Cassini-Soldner projection. 2 ρν b a (1 − e2 cos2 β)2 Helmert (1880. namely. Such a projection is only possible for surfaces of constant . The result is the familiar azimuthal equidistant projec= E(σ.7 Geodesic map projections sin σ1 cos σ2 J(σ2 ) − J(σ1 ) √ − . Eq. 18 (top sub-figure). φ1 . this gives the equidistant cylindrical projection. Necessary and sufficient conditions for a geodesic being the shortest path nents of the Jacobi field. §6) (Forsyth 1927. of The gnomonic projection is a projection of the sphere where all geodesics (i. ½π. we expect m12 to oscillate about zero. ik).1. Due to a sphere of radius 1/√K. θ) is centered on some point A. if the starting point of a geodesic is a pole. the supplemental condition m12 ≥ 0 is required if |λ12 | = π.e. then the reduced length is the radius of the circle of The rectangular coordinate system (x. Thus. Two map projections are defined in terms of geodesics. If this condition is not satisfied. m12 there is a nearby path (not necessarily a geodesic) which m12 is shorter. φ1 = while the scale in the azimuthal direction is s12 /m12 . are: The Gaussian curvature for an ellipsoid of revolution is • for an oblate ellipsoid. while the scale in the x direction is 1/M 32 . The scale of the projection in the y direcm12 = sin( Ks12 )/ K.)) solved the Gauss-Jacobi equation for this case obtaining • for a prolate ellipsoid. Cassini’s limit. terms of the reduced length (Karney 2013. |λ12 | ≤ π.geodesic being a global shortest path. the solutions for m12 and M 12 are the same as for map of France placed A at the Paris Observatory. |σ12 | ≤ π. Similarly. m12 = a cosβ2 = a sinσ12 . M31 = M32 M21 − (1 − M23 M32 ) m23 of the geodesic and is only a necessary condition for the The reduced length and the geodesic scale are compo. x = r cosθ and y = r sinθ. great circles) map to straight lines (making it a convenient aid to navigation). Due to the basic propof two geodesics starting at the same point with azimuths erties of geodesics (Gauss 1828). then M13 = M12 M23 − (1 − M12 M21 ) . y) uses a reference latitude. §§26–27) (Bliss 1916). To simplify the discussion of shortest paths in this paragraph we consider only geodesics with s12 > 0. in the field of the differential geometry of surfaces. 20 where the geodesics are numbered in order of increasing length. Carrying out this limit in the case of a general surface yields an azimuthal projection in which the distance from the center of projection is given by ρ = m12 /M 12 . so that there is no nearby path connecting the two points which is shorter. 20. Jacobi (1891) calls this star-like figure produced by the envelope an astroid. 19. The four geodesics connecting A and a point B. All the geodesics are tangent to the envelope which is shown in green in the figure. in parametric form. it is possible to construct an ellipsoidal gnomonic projection in which this property approximately holds (Karney 2013. The projection can then be used to give an iterative but rapidly converging method of solving some problems involving geodesics. §8). the gnomonic projection is the limit of a doubly azimuthal projection. points on the envelope may be computed by finding the point at which m12 = 0 on a geodesic (and Newton’s method can be used to find this point). Four such geodesics are shown in Fig. Here the geodesics for which α1 is a multiple of 3° are shown in light blue.) Some geodesic circles are shown in green. finding the intersection of two geodesics and finding the shortest path from a point to a geodesic. m12 > 0. thus there are two geodesics (with a length approximately half the circumference of the ellipsoid) between A and these points. The Hammer retroazimuthal projection is a variation of the azimuthal equidistant projection (Hammer 1910). (This figure uses the same position for A as Fig. Hinks (1929) suggested another application: if the central point A is a beacon. The envelope of geodesics from a point A at φ1 = −30°. 15 and is drawn in the same projection. Only the shortest line Geodesics from a single point (f = 1/10. The cut locus is shown in red. then at an unknown location B the range and the bearing to A can be measured and the projection can be used to Outside the astroid two geodesics intersect at each point. Table 1): The approximate shape of the astroid is given by x2/3 + y 2/3 = 1 or. The envelope is the locus of points which are conjugate to A. such as the Rugby Clock. all geodesics through the center of projection are straight. i. φ2 = 26°. A geodesic is constructed from a central point A to some other point B. 1 GEODESICS ON AN ELLIPSOID OF REVOLUTION 2 N 3 B 4 A 1 Fig. φ1 = −30°) (the first one) has σ12 ≤ π. 19. as B approaches A. However. The astroid is also the envelope of the family of lines . Fig. On the sphere.8 Envelope of geodesics The two shorter geodesics are stable. in particular.) 1.. Inside the astroid four geodesics intersect at each point. estimate the location of B.14 Gaussian curvature (Beltrami 1865). Thus a projection in which geodesics map to straight lines is not possible for an ellipsoid. the other two are unstable. λ12 = 175°. x = cos3 θ. A similar set of geodesics for the WGS84 ellipsoid is given in this table (Karney 2011. instead of at A). The geodesics from a particular point A if continued past the cut locus form an envelope illustrated in Fig. these form cusps on the envelope. Even though geodesics are only approximately straight in this projection. not for subsequent ones. The polar coordinates of the projection of B are r = s12 and θ = ½π − α2 (which depends on the azimuth at B. This corresponds to the situation on the sphere where there are “short” and “long” routes on a great circle between two points. (The geodesics are only shown for their first passage close to the antipodal point. a projection preserving the azimuths from two points A and B. This can be used to determine the direction from an arbitrary point to some fixed center.e. y = sin3 θ. C.e. cartographic projections uses the C implementation for Applying this formula to the quadrilateral AFHB. The library b2 2 = R22 Γ + − R 2 cos ϕ dϕ dλ. Implementations in several languages (C++. the PROJ. Now the Gauss–Bonnet theorem applied to a geodesic polygon states This result follows from one of Napier’s analogies. Multiplying the equation for Γ by R2 2 . Here we develop the formula for the area S 12 of AFHB The area of a geodesic polygon is given by summing S 12 following Sjöberg (2006). GeodSolve. where dT is an element of surface area and K is the 1 2 2 cos 2 (β2 − β1 ) Gaussian curvature. ∫ Γ= ∫ K dT = cos ϕ dϕ dλ. if it does 2π R2 must be added of the ellipsoid is to the sum. this method is accurate to about 0. The integral can be expressed as a series valid for small f (Danielsen 1989) (Karney 2013. where ) s .1 mm for the j. cos γ sin γ S12 = R22 (α2 −α1 )+b2 λ2 λ1 ( 1 tanh−1 (e sin ϕ) R22 + − 2 2e sin ϕ b 2(1 − e2 sin2 ϕ) where γ is a parameter.) This aids in finding a where the integral is over the geodesic line (so that φ is good starting guess for α1 for Newton’s method for in implicitly a function of λ). ) ∫ ( 1 JavaScript. Except for nearly anj tipodal points (where the inverse method fails to conis the geodesic excess and θj is the exterior angle at vertex verge). the geodesic circles are involutes S12 = R22 E12 −e2 a2 cos α0 sin α0 dσ.9. These are accurate to about 15 nanometers for the WGS84 ellipsoid. authalic radius. ∫ σ2 The astroid is the (exterior) evolute of the geodesic circles t(e′2 ) − t(k 2 sin2 σ) sin σ centered at A. Vincenty’s original formulas are used in many Γ = 2π − θj geographic information systems. §6 and addendum). 1 (Danielsen 1989).1. this li) ∫ ( brary can return m12 . 2 2 T = R2 Γ + − R2 cos ϕ dϕ dλ K In addition to solving the basic geodesic problem. and performing the integral over φ gives line utility. M 21 . Likewise. 2 e′2 − k 2 sin2 σ σ1 of the astroid. Once this area is known. and S 12 . x and the notation E 12 = α2 − α1 is used for the geodesic excess. t(x) = 1 + x + √ √ sinh−1 x 1+x √ . 1. Version 3. Converting this into an integral inverse problem in the case of nearly antipodal points over σ. Python. where 1. As of version 4. (These are generated by the rod of the trammel of Archimedes.10 Software implementations An implementation of Vincenty’s algorithm in Fortran is provided by NGS (2012). and in the library itself. Java. the area of a polygon may be computed by summing the contributions from all the edges of the polygon. Matlab. for geodesic 2 2 2 (1 − e sin ϕ) calculations. then a convenient expression for E 12 is ∫ ∫ 1 T = dT = cos ϕ dϕ dλ. and Maxima) are provided. i.0 includes Vincenty’s treatment of nearly antipodal points (Vincenty ∑ 1975b). This is exposed in the commandthat Γ = α2 − α1 . §5). Fortran. geod. the area of the quadrilateral AFHB in Fig. we obtain (Karney 2013. M 12 . where R2 is the WGS84 ellipsoid (Karney 2011.1.9 Area of a geodesic polygon A geodesic polygon is a polygon whose sides are geodesics. noting geodesic calculations. K sin 12 (β2 + β1 ) ω12 E12 tan tan = .. includes a command-line utility. This result holds provided2that the polygon does not include a pole. The area of any closed region over its edges. and subtracting this from the equation for The algorithms given in Karney (2013) are included in T gives[21] GeographicLib (Karney 2015). If the edges are specified by their vertices. The area of such a polygon may be found by first computing the area between a geodesic segment and the equator. §9).10 Software implementations 15 ∫ x y + = 1.4 library for where the value of K for an ellipsoid has been substituted. In a remarkable paper.01. §3.Y. √ a2 sin2 ω + b2 cos2 ω − c2 √ Z = c sin β . Z = c sin ϕ′ . ω) defined by Gaspard Monge √ a2 − b2 sin2 β − c2 cos2 β √ X = a cos ω . λ) are defined by   cos ϕ cos λ ∇h =  cos ϕ sin λ  . §§26–27) employed the ellipsoidal latitude and longitude (β. |∇h| sin ϕ The parametric latitude and longitude (φ′. a ≥ b ≥ c > 0. so the use of the symbol β is consistent by . et al. there was a complete understanding of the properties of geodesics on an ellipsoid of revolution. a b c (Klingenberg 1982.16 2 GEODESICS ON A TRIAXIAL ELLIPSOID The solution of the geodesic problems in terms of elliptic integrals is included in GeographicLib (in C++ only). without loss of generality. This method of solution is about 2–3 times slower than using series expansions. geodesics on a triaxial ellipsoid (with 3 unequal axes) have no obvious constant of the motion Charles Dupin and thus represented a challenging “unsolved” problem in the first half of the 19th century. β becomes the parametric latitude for an oblate ellipsoid.5).Z) are Cartesian coordinates centered on the ellipsoid and. addenda). Bessel. given by Clairaut’s relation allowing the problem to be reduced to quadrature. 2 Geodesics on a triaxial ellipsoid Solving the geodesic problem for an ellipsoid of revolution is. The geographical latitude and longitude (φ. λ′) are defined by X = a cos ϕ′ cos λ′ .[24] A point on the surface is specified by a latitude and longi2. Jacobi (1839) discovered a constant of the motion X2 Y2 Z2 allowing this problem to be reduced to quadrature also h = 2 + 2 + 2 = 1.. Consider the ellipsoid defined In the limit b → a. e.).g. 100] (Karney 2013. Oriani. geodesics have a constant of the motion.1 Triaxial coordinate systems tude. By the early 19th century (with the work of Legendre. however it provides accurate solutions for ellipsoids of revolution with b/a ∈ [0. relatively simple: because of symmetry. a2 − c2 Y = b cos β sin ω. from the mathematical point of view. Jacobi (1866. via the -E option to GeodSolve. Y = b cos ϕ′ sin λ′ .[22][23] where (X. a2 − c2 The key to the solution is expressing the problem in the “right” coordinate system. On the other hand. Part 5). Y = 0.8.[25] 1 dα 1 = × 2 2 2 ds ((a − b ) sin ω + (b2 − c2 ) cos2 β)3/2 √ ( 2 (a − b2 ) cos ω sin ω a2 sin2 ω + b2 cos2 ω − c2 √ cos α a2 sin2 ω + b2 cos2 ω √ ) (b2 − c2 ) cos β sin β a2 − b2 sin2 β − c2 cos2 β √ sin α . Finally they are geodesic ellipses and hyperbolas defined using two adjacent umbilical points (Hilbert & Cohn-Vossen 1952. Jacobi showed that the geodesic equations. Jacobi’s solution The grid lines of the ellipsoidal coordinates may be interpreted in three different ways 1. expressed in ellipsoidal coordinates. 188). 21 can be generated with the C. + b2 sin2 β + c2 cos2 β Grid lines of constant β (in blue) and ω (in green) are given in Fig.2. ω) is an orthogonal coordinate system: the grid lines intersect at right angles. The third principal section. (β.01:1:0. are separable. they are parallel to the directions of principal curvature (Monge 1796). 2. λ = 30°. i. 21.(Jacobi 1839. However. G. Here is how he reConversions between these three types of latitudes and counted his discovery to his friend and neighbor Bessel longitudes and the Cartesian coordinates are simple alge. with the previous sections. the lines of constant β in Fig. and it is viewed in an orthographic projection from a point above φ = 40°. ω is different from the spherical longitude defined above. Jacobi familiar string construction for ellipses with the ends of the string pinned to the two umbilical points. Letter to Bessel). They are “lines of curvature” on the ellipsoid. Ellipsoidal coordinates. I reduced to quadrature the problem of geodesic lines on . For example. Here and in the other figures in this section the parameters of the ellipsoid are a:b:c = 1. 2. λ′). 21..2 Jacobi’s solution 17 ds2 b2 sin2 β + c2 cos2 β = dβ 2 (a2 − b2 ) sin2 ω + (b2 − c2 ) cos2 β a2 − b2 sin2 β − c2 cos2 β + a2 sin2 ω + b2 cos2 ω dω a2 sin2 ω + b2 cos2 ω − c2 and the differential equations for a geodesic are √ a2 − b2 sin2 β − c2 cos2 β √ b2 sin2 β + c2 cos2 β (a2 − b2 ) sin2 ω + (b2 − c2 ) cos2 β √ dω 1 a2 sin2 ω + b2 cos2 ω − c √ =√ ds a2 sin2 ω + b2 cos2 ω (a2 − b2 ) sin2 ω + (b2 − c2 ) cos2 β dβ =√ ds Fig. p.e. is covered by the lines β = ±90° and ω = 0° or ±180°. The element of length on the ellipsoid in ellipsoidal coordinates is given by The day before yesterday. braic exercises. 3. The principal sections of the ellipsoid. In contrast to (φ. J.2 defined by X = 0 and Z = 0 are shown in red. They are also intersections of the ellipsoid with confocal systems of hyperboloids of one and two sheets (Dupin 1813. λ) and (φ′. These lines meet at four umbilical points (two of which are visible in this figure) where the principal radii of curvature are equal. An alternative expression for the distance is √ √ b2 sin2 β + c2 cos2 β (b2 − c2 ) cos2 β − γ dβ √ ds = a2 − b2 sin2 β − c2 cos2 β √ √ a2 sin2 ω + b2 cos2 ω (a2 − b2 ) sin2 ω + γ dω √ + . The solution given by Jacobi (Jacobi 1839) (Jacobi 1866. is found using √ ds b2 sin2 β + c2 cos2 √ = √ 2 (a2 − b2 ) sin ω + (b2 − c2 ) cos2 β a2 − b2 sin2 β − c2 cos2 β (b2 √ a2 sin2 ω + b2 cos2 √ =√ a2 sin2 ω + b2 cos2 ω − c2 (a J.. The constant γ may be expressed as Joseph Liouville γ = (b2 − c2 ) cos2 β sin2 α − (a2 − b2 ) sin2 ω cos2 α. s. These two functions are just Abelian integrals. However..” Two constants δ and γ appear in the solution. §§583–584) where he gives the solution found by Liouville (1846) for general quadratic surfaces. '38. which become the well known elliptic integrals if 2 axes are set equal.18 2 GEODESICS ON A TRIAXIAL ELLIPSOID §28) is √ b2 sin2 β + c2 cos2 β dβ √ δ= √ a2 − b2 sin2 β − c2 cos2 β (b2 − c2 ) cos2 β − γ √ ∫ a2 sin2 ω + b2 cos2 ω dω √ − . Abelian integrals. Königsberg. 28th Dec. Typically δ is zero if the lower limits of the integrals are taken to be the starting point of the geodesic and the direction of the geodesics is determined by γ. In the limit b → a. √ a2 sin2 ω + b2 cos2 ω − c2 (a2 − b2 ) sin2 ω + γ ∫ As Jacobi notes “a function of the angle β equals a function of the angle ω. Darboux an ellipsoid with three unequal axes.. In this formulation. we have γ = 0 and δ determines the direction at the umbilical point. this reduces to sinα cosβ = const. a2 sin2 ω + b2 cos2 ω − c2 . They are the simplest formulas in the world. the distance along the geodesic. A nice derivation of Jacobi’s result is given by Darboux (1894. G. the familiar Clairaut relation. for geodesics that start at an umbilical points. where α is the angle the geodesic makes with lines of constant ω. and Z = 0. ω1 = 0° (an umbilical point). 22–26. if the starting point is at a higher latitude (Fig. on each oscillation it completes slightly more that a full circuit around the ellipsoid resulting. there are only 3 simple closed geodesics. it is convenient to consider geodesics which intersect the middle principal section. If the starting point is β1 = 90°. Fig. On a triaxial ellipsoid. in the geodesic filling the area bounded by the two longitude lines ω = ω1 and ω = 180° − ω1 . Note that a single geodesic does not fill an area on the ellipsoid. ω1 = 0. ω1 = 0°. then γ < 0 and the geodesic encircles the ellipsoid in a “transpolar” sense. 26. All tangents to umbilical geodesics touch the confocal hyperbola which intersects the ellipsoid at . 223–224). in the typical case.966°. 22) the distortions resulting from a ≠ b are evident. then γ = 0 and the geodesic repeatedly intersects the opposite umbilical point and returns to its starting point. 24. compare to Fig. 180°). However. Fig. which use the same ellipsoid parameters and the same viewing direction as Fig. 21. β1 = 90°. Two examples are given in Figs. Such geodesics are shown in Figs. This is shown in Fig. The geodesic oscillates east and west of the ellipse X = 0.2. ω1 = 9. 90°). pp. 25. Y = 0. 22 and 23. 12 (rotated on its side).[26] To survey the other geodesics. If a = b. p. β1 = 45. in this case. and α1 = 180°. Y = 0. in the geodesic filling the area bounded by the two latitude lines β = ±β1 .1°. all meridians are geodesics. 24 and 25. The constriction of the geodesic near the pole disappears in the limit b → c. Figure 22 shows practically the same behavior as for an oblate ellipsoid of revolution (because a ≈ b). β1 = 87. Circumpolar geodesics.3 Survey of triaxial geodesics Survey of triaxial geodesics 19 Transpolar geodesics. 24 would resemble Fig. 265). Fig. on each circuit the angle at which it intersects Y = 0 becomes closer to 0° or 180° so that asymptotically the geodesic lies on the ellipse Y = 0 (Hart 1849) (Arnold 1989. In addition. Fig. Two examples are given in Figs. All tangents to a transpolar geodesic touch the confocal double-sheeted hyperboloid which intersects the ellipsoid at ω = ω1 . ω1 = 39. in the typical case. the three principal ellipses are shown in red in each of these figures.9°. and α1 = 135° (the geodesic leaves the ellipse Y = 0 at right angles). then γ > 0 and the geodesic encircles the ellipsoid in a “circumpolar” sense. However.48°. The geodesic oscillates north and south of the equator. ω1 ∈ (0°. and α1 = 90°. If the starting point is β1 = 90°. the three principal sections of the ellipsoid given by X = 0. the effect of a ≠ b causes such geodesics to oscillate east and west. the ellipsoid becomes a prolate ellipsoid and Fig. If the starting point is β1 ∈ (−90°. All tangents to a circumpolar geodesic touch the confocal singlesheeted hyperboloid which intersects the ellipsoid at β = β1 (Chasles 1846) (Hilbert & Cohn-Vossen 1952. α1 = 90°. at right angles. 23. 22. 11. on each oscillation it completes slightly less that a full circuit around the ellipsoid resulting. α1 = 180°.3 2. ) 3 Applications The direct and inverse geodesic problems no longer play Henri Poincaré the central role in geodesy that they once did. α1 = 135°. If it is perturbed. ω1 = 0°. the closed geodesic on the ellipse Y = 0. terrestrial problem are now solved by three-dimensional methods geodesics still play an important role in several areas: . there are two umbilical geodesics. Nevertheless. these (Vincenty & Bowring 1978). • Whereas the closed geodesics on the ellipses X = 0 and Z = 0 are stable (an geodesic initially close to and nearly parallel to the ellipse remains close to the ellipse). Instead of solving adjustment of geodetic networks as a twodimensional problem in spheroidal trigonometry. which goes through all 4 umbilical points. • Through any point on the ellipsoid. • The geodesic distance between opposite umbilical points is the same regardless of the initial direction Karl Weierstrass of the geodesic. An umbilical geodesic. then its envelope is an astroid with two cusps lying on β = −β1 and the other two on ω = ω1 + π (Sinclair 2003). 26. it does not utilize Jacobi’s solution. is exponentially unstable. (This behavior may repeat depending on the nature of the initial perturbation. β1 = 90°. Umbilical geodesic enjoy several interesting properties. (Panou 2013) gives a method for solving the inverse problem for a triaxial ellipsoid by directly integrating the system of ordinary differential equations for a geodesic. The cut locus for A is the portion of the line β = −β1 between the cusps (Itoh & Kiyohara 2004).) If the starting point A of a geodesic is not an umbilical point. the umbilic points. it will swing out of the plane Y = 0 and flip around before returning to close to the plane.20 3 APPLICATIONS Fig. (Thus. the element of distance on the ellipsoid is given by ds2 = (a2 sin2 β2 + b2 cos2 β) dβ2 + a2 cos2 β dλ2 . [6] In terms of β. Bessel was unaware.21 • for measuring distances and areas in geographic information systems. expanding in a Taylor series. of which. Levallois & Dupuy (1952) gave recurrence relations for the series in terms of Wallis’ integrals and Pittman (1986) describes a similar method. . integration. [8] Other choices of independent parameter are possible. For this reason. This effort failed. this is given by α2 ± π. [9] Nowadays. (6) and (7) are real. Thus. e. • extensions to an arbitrary number of dimensions (Knörrer 1980). • geodesic flow on a surface (Berger 2010.g. 12). • methods for solving systems of differential equations by a change of independent variables (Jacobi 1839). [7] It may be useful to impose the restriction that the surface have a positive curvature everywhere so that the latitude be single valued function of Z. [2] This prompted a courteous note by Oriani (1826) noting his previous work. expressions in terms of elliptic integrals. Earlier. see also Forsyth (1927. many problems in physics can be formulated as a variational problem similar to that for geodesics.) [14] When solving for σ. but equivalent. • the definition of maritime boundaries (UNCLOS 2006). Art. p. and performing trigonometric simplifications. • investigations into the number and stability of periodic orbits (Poincaré 1905). [11] Despite the presence of i = √−1. Clairaut’s relation is just a consequence of conservation of angular momentum for a particle on a surface of revolution. §8. using the atan2 function. presumably. the elliptic integrals in Eqs. or ω using a formula for its tangent. Some authors calculate the back azimuth instead.06). this refers to Cassini’s proposed map projection for France (Cassini 1735) where one of the coordinates was the distance from the Paris meridian. Indeed. can be carrying using a computer algebra system. the quadrant should be determined from the signs of the numerator of the expression for the tangent. but otherwise subject to no forces (Laplace 1799a) (Hilbert & Cohn-Vossen 1952. α. 5 Notes [1] Here α2 is the forward azimuth at B. but this is not suitable for large distances. §75. • in the rules of the Federal Aviation Administration for area navigation (RNAV 2007). [10] Legendre (1806. [3] Clairaut (1735) uses the circumlocution “perpendiculars to the meridian". • the study of caustics (Jacobi 1891). • in the limit c → 0. • the development of differential geometry (Gauss 1828) (Christoffel 1869). A similar proof is given by Bomford (1952. which resulted in an angry rebuttal by Bessel (1827). In particular many authors use the vertex of a geodesic (the point of maximum latitude) as the origin for σ. 13) also gives a series for σ in terms of s. By the principle of least action. geodesics on simple surfaces such as ellipsoids of revolution or triaxial ellipsoids are frequently used as “test cases” for exploring new methods. [13] It is also possible to express the integrals in terms of Jacobi elliptic functions (Jacobi 1855) (Luther 1855) (Forsyth 1896) (Thomas 1970. Appendix 1). and also a thinly veiled accusation of plagiarism from Ivory (1826) (his phrase was “second-hand from Germany”). Chap. [4] Kummell (1883) attempted to introduce the word “brachisthode” for geodesic. Halphen (1888) gives the solution for the complex quantities R exp(±iλ) = X ± iY in terms of Weierstrass sigma and zeta functions. the geodesic problem is equivalent to the motion of a particle constrained to move on the surface. 4 See also • Geographical distance • Great-circle navigation • Geodesics • Geodesy • Meridian arc • Rhumb line • Vincenty’s formulae [12] Rollins (2010) obtains different. Examples include: • the development of elliptic integrals (Legendre 1811) and elliptic functions (Weierstrass 1861). [5] Laplace (1799a) showed that a particle constrained to move on a surface but otherwise subject to no forces moves along a geodesic for that surface. geodesics on a triaxial ellipsoid reduce to a case of dynamical billiards.. • the method of measuring distances in the FAI Sporting Code (FAI 2013). 222). the necessary algebraic manipulations. This form is of interest because the separate periods of latitude and longitude of the geodesic are captured in a single doubly periodic function. 5. Numerical Algorithms 10 (1): 13–98. doi:10. [17] Bessel (1825) treated the longitude integral approximately in order to reduce the number of parameters in the equation from two to one so that it could be tabulated conveniently. W. [22] This section is adapted from the documentation for GeographicLib (Karney 2015. (1735).1007/BF03198517. ISBN 978-0-9802327-3-8. W. this leads to a singular integrand (Karney 2011. Chapter 11. [21] Sjöberg (2006) multiplies Γ by b2 instead of R2 2 . In Proceedings 1991 International Aerospace and Ground Conference on Lightning and Static Electricity. C. Philosophical Magazine (4th ser. Errata.. geodesics on a surface of the complexity of the geoid are partly chaotic (Waters 2011). G. Mathematical Methods of Classical Mechanics. “The calculation of longitude and latitude from geodesic measurements”. [18] If φ1 = φ2 = 0. F. 6 References • Arnold. Translated by C. “Risoluzione del problema: Riportare i punti di una superficie sopra un piano in modo che le linee geodetiche vengano rappresentate da linee rette” [Mapping a surface to a plane so that geodesics are represented by straight lines].) . Nachr. Senechal. “On the geodesic lines on an oblate spheroid”. G. Air Force (FTDMT-64-390) REFERENCES • Beltrami. E. (1967) [1962]. (2010). (2010) [1825]. where a and b stood for the equatorial radius and polar semi-axis. • Carlson. Annali di Matematica Pura ed Applicata. Translated by K. §17) uses the term “coefficient of convergence of ordinates” for the geodesic scale. • Bessel.177B. Strang. Transactions of the American Mathematical Society 17 (2): 195–206. 2). J. R. “Numerical computation of real or complex elliptic integrals”. ω changes quadrants in step with σ. doi:10. • Bomford. Deakin. “De la carte de la France et de la perpendiculaire a la méridienne de Paris” [The map of France and the perpendicular to the meridian of Paris]. Oxford: Clarendon. • Berger.1007/BF02198293. (1870). OCLC 1396190.. I. E. [25] The limit b → c gives a prolate ellipsoid with ω playing the role of the parametric latitude. (1952).). Astronomische Nachrichten (in German) 5 (108): 177–180. F.19). Karney & R. Thus the corresponding inequalities are a = a ≥ b > 0 for an oblate ellipsoid and b ≥ a = a > 0 for a prolate ellipsoid.1824. Geodesy. [16] Because tanω = sinα0 tanσ.1007/978-3540-70997-8. Springer. F. 13 and 14 (Klingenberg 1982. doi:10. (1916). K.18270051202. Astronomische Nachrichten 331 (8): 852–861. Bent.201011352. [19] The ordering in relations (8) automatically results in σ12 > 0. Weinstein (2nd ed. OCLC 795014501. It is therefore straightforward to express λ2 so that λ12 indicates how often and in what sense the geodesic has encircled the ellipsoid.1002/asna. 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B. • Implementation of Karney (2013) for ellipsoids of revolution in Geographiclib (Karney 2015): • GeographicLib web site for downloading library and documentation. (2010).10504515. Experimental Mathematics 12 (4): 477–485. • Vincenty. doi:10. • Implementations of Vincenty (1975a) for oblate ellipsoids: • NGS implementation. T.1061/(ASCE)SU. (1861). man page for a utility for calculating the area of geodesic polygons. M. (2012). includes modifications described in Vincenty (1975b). (1978). R. 7 EXTERNAL LINKS • Waters. • GeodSolve(1). “Direct and inverse solutions of geodesics on the ellipsoid with application of nested equations” (PDF).1179/003962609X451663. sphere and ellipsoid”. “On the last geometric statement of Jacobi”. “The computation of long geodesics on the ellipsoid by non-series expanding procedure”. SP-138. doi:10.23.1943-5428.0000061. • Javascript implementations of solutions to direct problem and inverse problem. doi:10. Bibcode:2012PhyD.1016/j. • Thomas. • Saito. • geod(1). • The first description of the geodesic algorithms from (Karney 2009).. • Weierstrass. • Sjöberg. A Manual on Technical Aspects of the United Nations Convention on the Law of the Sea.53.1179/sre.010. (1979).176. Bulletin Géodésique 98: 341–373. • Drawing geodesics on Google Maps. T. Shirazian. NOS NGS-13. U.. NOAA. (2003). T. doi:10. (2005). • Saito.2011. ena 241 (5): 543–552. • An online version of GeodSolve. doi:10. W. man page for a utility for geodesic calculations. "Über die geodätischen Linien auf dem dreiaxigen Ellipsoid” [Geodesic lines on a triaxial ellipsoid]. • Sjöberg.1007/BF02522166. Bibcode:1979BGeod. Bibcode:1970BGeod.1975. P. L. • An online version of Planimeter. Geodetic inverse solution between antipodal points (PDF) (Technical report). Naval Oceanographic Office. • Matlab implementation of the geodesic routines (used for the figures for geodesics on ellipsoids of revolution in this article). (1970). “Determination of areas on the plane. • UNCLOS (2006). “The computation of long geodesics on the ellipsoid through Gaussian quadrature”. Monaco: International Hydrographic Bureau. D.543W. M. Monatsbericht Königlichen Akademie der Wissenschaft zu Berlin (in German): 986–997. .). C. doi:10. “An integral for geodesic length” (PDF). approximately 180 books and articles on geodesics on ellipsoids together with links to online copies. • Geodesics on a triaxial ellipsoid: • Additional notes about Jacobi’s solution. T. 1982 (PDF) (Technical report) (4th ed. E. Journal of Surveying Engineering 131 (1): 20–26... • Vincenty. E. Survey Review 42 (315): 20–26. Journal of Geodesy 53 (2): 165–177. T.3231.88. E.1080/10586458. doi:10. Retrieved 2011-07-28.11. R. & Local Geometry (Technical report). Bowring. • Online calculator from Geoscience Australia. “Solving the direct and inverse geodetic problems on the ellipsoid by numerical integration”.S. Addendum: Survey Review 23 (180): 294 (1976)..1061/(ASCE)0733-9453(2005)131:1(20). “Regular and irregular geodesics on spherical harmonic surfaces”. Spheroidal Geodesics.1179/003962606780732100. K. • Planimeter(1). T. • Sinclair.2003. doi:10. Reference Systems.1007/BF02521087... C. M. Physica D: Nonlinear PhenomarXiv:1112. DMAAC Geodetic Survey Squadron. (1970). (1975a). man page for the PROJ.. • Vincenty. • Caustics on an ellipsoid. • Thomas. PDF. L.. Journal of Surveying Engineering 138 (1): 9–16.241. Application of three-dimensional geodesy to adjustments of horizontal networks (PDF) (Technical report). • Javascript utility for direct and inverse problems and area calculations. Survey Review 38 (301): 583–593.4 utility for geodesic calculations. T. (2006). doi:10. Featherstone. J.165S. (1975b).44.physd. 7 External links • Online geodesic bibliography. Survey Review 23 (176): 88–93. Fgnievinski.svg License: CC BY-SA 3.svg License: CC BY-SA 3. Scotland) Original artist: User MIstvan on hu.wikimedia.0 Contributors: Own work Original artist: Cffk • File:Differential_element_of_a_meridian_ellipse.wikimedia.0 Contributors: Own work Original artist: Cffk • File:Darboux.wikimedia.svg License: CC BY-SA 3. JuanFox.wikimedia. DEMcAdams. David Eppstein.si.jpg License: Public domain Contributors: Gauß-Gesellschaft Göttingen e.0 Contributors: Own work Original artist: Cffk • File:F-R_Helmert_1.st-and. Crapscourge. Arthur Rubin.jpg Source: https://upload.svg Source: https://upload.wikimedia. Bld Bonne-Nouvelle.svg Source: https://upload.wikimedia.org/wikipedia/commons/a/a9/Arthur_Cayley. 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